Portfolio Risk Optimizer Target Return

Mean-variance optimization describes an asset by two numbers, expected return and standard deviation of return, and describes a portfolio by the weighted mix of those plus the correlation between the assets. For two assets the portfolio variance is w1^2 s1^2 + w2^2 s2^2 + 2 w1 w2 s1 s2 rho, and it is the cross term that makes diversification work: whenever correlation rho is below 1, combined volatility falls below the weighted average of the parts. The stock weight needed for a target return is a straight interpolation, (R_target - R_bond) divided by (R_stock - R_bond), so a target above the stock return or below the bond return cannot be reached without leverage or short positions. Typical long-run US inputs are roughly 10 percent nominal return with 15 to 18 percent volatility for large-cap equity, 4 to 5 percent with 5 to 7 percent volatility for investment-grade bonds, and a stock-bond correlation that has swung from strongly negative in the 2000s to clearly positive in 2022. The Sharpe ratio, excess return divided by volatility, sits between about 0.3 and 0.8 for most diversified portfolios over long horizons. The parametric 95 percent value at risk used here, expected return minus 1.645 standard deviations, assumes normally distributed returns and therefore understates real drawdowns, because equity returns have fat tails and correlations tend to rise precisely in the crises when diversification is most needed. Estimation error in expected returns is the larger practical weakness: small changes in assumed returns swing the optimal weights sharply.

Harry Markowitz published Portfolio Selection in the Journal of Finance in March 1952, written while he was a doctoral student at the University of Chicago. Its contribution was to replace the then-common goal of maximizing expected return with an explicit trade-off between expected return and variance, and to show that portfolio risk depends on covariances between holdings rather than on the riskiness of each holding considered alone. The efficient frontier and the mean-variance optimization problem both come from that paper and its 1959 book-length expansion. James Tobin added the separation theorem in 1958, showing that once a risk-free asset exists every investor holds the same risky tangency portfolio and adjusts risk only through the amount lent or borrowed, which is the tangency calculation reported here. William Sharpe developed the Capital Asset Pricing Model in 1964 and introduced the reward-to-variability ratio in 1966, later renamed the Sharpe ratio. Markowitz, Sharpe and Merton Miller shared the 1990 Nobel Memorial Prize in Economic Sciences for this body of work. Value at Risk entered mainstream practice through JP Morgan, which published the RiskMetrics methodology openly in 1994, after which the measure was written into the Basel market-risk capital rules and became a standard reporting figure for banks and asset managers.